lnat. J. Math. a Math.
Vol.
(1978) 203-208
203
A NOTE ON STRICTLY CYCLIC SHIFTS ON
p
GERD H. FRICKE
Department of Mathematics
Wright State University
Dayton, Ohio 45431
(Received June 28, 1977)
ABSTRACT.
In this paper the author shows that a well known sufficient
condition for strict cycliclty of a weighted shift on
I
P
is not a necessary
condition for any p with I < p < =.
i.
INTRODUCTION.
For 1 < p <
let
p
be the Banach space of absolutely p-summable se-
Let S
quences of complex numbers.
welght sequence
and
Sn
[2]
and
u
ulu...u+/z
n
{
denote the weighted shift on
}
n=n
defined
for all n
_>
i
u
bY S
0Xn
en]
n=Elnxn-len"
p
with
Let
0
i
(For more detail we refer the reader to
[3]).
Mary Embry [i] showed that for p
i the weighted shift S
is strictly cyclic
if and only if
sup
n,m
n+m
<
(R).
(.. )
G.H. FRICKE
204
[2]
Edward Kerlin and Alan Lambert
for the case 1 < p <
1
with
P
+
considered the natural extension to (i.i)
i
q
n
E
m=0
sup
n
and showed (1.2) implies S
is strictly cyclic on
a
(.2)
<
They also proved that
p
(1.2) is necessary if the weight sequence a is eventually decreasing.
(1.2)
This strongly suggested that
strict cyclicity.
However, we will show in this paper that (1.2) is not a
necessary condition for S
2.
is a necessary and sufficient condition for
.
to be strictly cyclic for any p with i < p <
PROOF.
To preserve the clarity of the proof we will consider the cases i < p
and 2 < p <
i
+ qi i. Let
Let i < p < 2 and let q be such that
p
sequence of rapidly increasing positive integers, e.g., cho6se nI
-l(nk)-
(10%_ 1 )
oq%_
be a
I0 and
for k > 1.
{a
We now define the weight sequence
i
by
I
an I
a2
i and for
k>l
-I
Clearly,
ifnk_
<
if n
k
nk_ 1
!nk-nk_
< i
<_nk.
ank_nk_ I and ank_Z+l_< a for
ank_l+l ank_l+2
Thus, for 0 < m < nk
3mBnk_m
2
separately.
(a)
n
k
<_
a a
l 2
cz
>
>
m
ala2...a nk_ 1
(= n )
k
1 <
<
T"
205
SRICTLY CYCLIC SHIFTS
Therefore,
nk-i
E
m=l
8nk
8m
8nk
q
nk (ank
>
m
-qnk-I
qnk_ 1
nknk--1
as k
/
/
Henc e,
We now show that S
sup
n
l
n
m=0
8n
8mSn_m
[’q
It is known that S [2] is strictly
is strictly cyclic.
cyclic if and only if
n
l
m=0
8n
for all x,y e
<
8mSn-m XmYn_m
(1.3)
P
Obviously, for 0 < m < n
an-m+I
8n
8mSn_m
n
and
i for m
a
a
ala 2
0 or m
n.
[p
Z {
n=0
n <
a
m
Thus
==in
Z
n=0
7.
m=0
8n
8mBn-m
XmYn_m
n-i
<
x0Yn
+
Y0Xn
+ Z
m=l
n
XmYn_ml } p
}P
<
n--O
since a,x,y
Hence (1.3) holds and S
(b)
Let 2 < p <
is strictly cyclic.
and iet
[+
increasing integers, e.g., choose n
nk
l
n=l
P
1.n
>
i.
1
(lOnk-i)
Let {_
be a sequence og rapidly
iO and n such that
k
lOnk_ 1
fork> i.
206
n
Define
and s
{dt} 1
2Sk_ I +
k
g d
i=l i
for k > i.
such that
2nk
G.H. FRICKE
1
q
for n
n
1,2,’.-and define
We now define the weight sequence
2
d
{i}l
Sk-Sk_l-i+l
by
Sk_ I
if
Sk_l+nk
if
s
k
s1
1
< i
if
-Sk_ I
{Sk} 1
by s 1
10
1 and for k > i,
<_ Sk_l+nk
< i
<_ s k
Sk_ I
<_ s k.
< i
s
Now, for Sk_ 1 < m
8Sk
<-k
sk_m+l
8mSsk_
Sk
a
elS2
m
sk_m+1
>
m
sSk-Sk_l
Sk_l+l
-2Sk_ I
m
-2Sk_ m-sk_ I
>__ nk_ I I
i=l
-2Sk_ 1
nk_ 1
k
-2qsk-i
>_nk_ I
m=0
Henc e,
nk
l i
i=l
-i 0nk-i nk
l
>_ nk_ I
-I
i
-I
/
as k
i=l
sup
n
l
n
m--O
We now show that
n
n=O
If
Sk_ 1
< n < s
m=0
k
8
8mSn_m XmYn-m
Sk_ 1
<
for all x,y e
and 0 < m < n then,
P
mn_m < nk
/
i
1
(m-sk_I)
Thus,
s
d
(R).
STRICTLY CYCLIC SHIFTS
mln{m,n-m} then, for s k
Let h
Sk_ I <_ n <_
s
k
207
and 0 < m < n,
I
Sn
88
m n-m
i
Thus,
Sk-Sk_l-i
Z
k=2
Z
n=Sk_l+l
Let
then
P
8n
n-i
l
m=l
E
k--2
8mSn-m XmYn_m
2
+--=
> q and
P
i.
E
n--Sk_l+l
n
2
r.
m=l
]XmYn_ml
Ilxll P
Let M
m=l
/
I1 11
Then,
s
k=2
8
nl 8mSn-m
XmYn-m
k
n
m=l
n=sk-Sk_ I
s
kffi2
n--sk-Sk_ 1
s
<
k
m=l
i
h- l XmYn_m
mln{m, n-m}
where h
k
l
l
k=2
n--sk-Sk_ 1
Combining (1.4) and (1.5) we obtain that (1.3) is satisfied.
QED
References
i.
Embry, Mary. Strictly Cyclic Operator Algebras on a Banach Space, to appear
in Pac. J. Math.
2.
Kerlln, Edward and Alan Lambert.
Math. 35 (1973) 87-94.
3.
Lambert, Alan. Strictly Cyclic Weighted Shifts, Proc. Amer. Math. Soc. 29
(1971) 331-336.
Strictly Cyclic Shifts on
Z
P’
Acta Scl.
.
208
.
H. FICKE
KEY (ORPS ANP PHRASES. (t c(t 1., <zbaot.u,te
AlS(kiOS) SLIB.TECT CLASSIFICATION [1970} CO/)F_.S.
40H05.
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